diffcb 0.1.6

Differentiable Critical Bandwidth: Silverman's modality test as a differentiable PyTorch layer with IFT backward pass.

DCB — Differentiable Critical Bandwidth

A PyTorch package that makes Silverman's critical bandwidth test (1981) fully differentiable, enabling end-to-end gradient-based optimization over the modal structure of continuous distributions.

Overview

The critical bandwidth h_crit is the minimum KDE bandwidth at which a distribution appears to have at most m modes — a classical nonparametric statistic for modality testing. DCB replaces every non-differentiable operation in its computation with a smooth surrogate, then uses the Implicit Function Theorem to compute exact gradients through the root-finding step at O(1) memory cost.

import torch
from dcb import DCBLayer

X = torch.randn(1000, requires_grad=True)   # 1D samples
layer = DCBLayer(target_modes=1)
h_crit = layer(X)                           # differentiable scalar
h_crit.backward()                           # exact IFT gradients

Installation

pip install diffcb

Or from source:

git clone https://github.com/ryZhangHason/differentiable-critical-bandwidth
cd differentiable-critical-bandwidth
pip install -e ".[dev]"

Accuracy vs R's bw.crit

DCB is validated against R's multimode::bw.crit(data, mod0=1) — the standard reference implementation of Hall & York (2001). On identical data:

n	DCB vs R (same sample)	DCB vs R (independent samples)
100K	0.004%	~0.5% (MC noise from independent RNG)
1M	0.005%	~0.2%
10M	0.004%	~0.1%

The independent-sample figures reflect natural sampling variability (two unbiased estimators drawing different data), not algorithmic error. On identical data, DCB agrees with R to within 0.005% at all tested n. DCB is 43× faster than R at n=100M (1.1 s vs 50 s) and handles n=2B in 24 s while R OOMs.

Key Parameters

DCBLayer(
    target_modes=1,       # target number of modes
    G=512,                # IFT evaluation grid points
    use_fft=True,         # FFT forward (default); eliminates subsampling bias for n>50K
    max_n_exact=1_000_000,# sketch to sketch_size when n exceeds this (None = always exact)
    sketch_size=500_000,  # sketch target; 500K matches full-n accuracy (O(n^{-2/9}) rate)
    safe_backward=False,  # clamp IFT denominator near bifurcations
)

Confirmed Experimental Results

All GPU results produced on Kaggle (T4 / P100) — see experiments/ and outputs/.

Experiment	Result	Criterion
Accuracy vs R (same data, n=100K)	0.004%	< 0.01% ✓
Validation (m≥2, Marron-Wand)	R²=0.91, MAE=0.07, ρ=0.89	R²≥0.85 ✓
Speedup vs scipy (CUDA T4, n=8192)	10.5×	≥3× ✓
GAN mode preservation	h_crit=1.232 >> 0.3	h_crit>0.3 ✓
Anomaly AUC (KDDCup99)	DCB=0.9982 vs IF=0.9867	DCB≥IF ✓

Repository Structure

dcb/            Core PyTorch package
  layer.py        DCBLayer nn.Module + DCBFunction autograd
  solver.py       IFT root-finder and backward pass
  fft_kde.py      FFT-based mode counter (MPS-safe, float64, G=16384)
  kde.py          Direct KDE derivatives (small-n path)
  utils.py        Grid, Silverman bandwidth, sg() stabilizer
experiments/    Reproduction scripts for all paper figures and tables
  phase1_*.py     Validation, speedup, ablation (Figures 1–2, S1–S2)
  phase2_gan.py   GAN mode-collapse prevention (Figure 3)
  phase3_anomaly.py  Anomaly detection (Table 2, Figure 5)
  round20_*.py    Large-n R comparison and streaming benchmarks
  round21_*.py    Accuracy improvement experiments
tests/          Unit tests (pytest, 45 passed, 1 xfailed)
outputs/        All generated figures and tables (PDFs, PNGs, CSVs)

License

Apache 2.0 — see LICENSE.